Chaosheng Zhang wrote: [snip]
Hi Chaosheng: A few points about log transforms. See below. > The reason why I care about this issue is that there are at least two > problems related to data transformation (in order to follow the normal > distribution): > > (1) The measurement scale is reduced. The orignal ratio/interval scale may > be reduced to the lower level of ordinal, even close to nominal, which > results in loss of raw information. There shouldn't be any loss of information since the log transformation is a one-to-one mapping. The sample variance is much smaller in the log scale but the log transform is often used precisely for that purpose. > (2) Artificial relationship is introduced. We know that the lognormal > distribution is widely accepted. In correlation analysis, if the > log-transformed data are used, the correlation becomes the "log-log" > relationship, not the oginal linear relationship. In bivariate regression > analysis, the original function is: > y = a x + b > However, for the log-transformed data, the function becomes: > log(y) = a log(x) + b > or y = exp (a log(x) + b) > In many cases, it is not clear if the relationship should be linear or > "log-linear". However, the artificially introduced "log-linear" relationship > need to be proved. Rather, when you apply the log transform you a-priori assume the existence of what you call the 'artificial relation', and this assumption refer to the algebraic form of the error term rather than to the relation between y and x. Say E(y)=f(x) is the model for y versus x, where E is the expectation operator. If you assume an additive error structure y_i=f(x_i)+e_i, where i indexes observation, and you consider the e_i's as iid normal random variates, then there is no reason to apply the log tranform. On the other hand if you assume a multiplicative error structure such as y_i=f(x_i)*e_i, and you assume that the e_i are iid lognormal random variates, then the log transform yields ln(y_i)=lnf(x_i)+ln(e_i) and now the ln(e_i) are iid normal random variates (with a much smaller variance than the e_i). The theory of the lognormal is well developed so that there isn't actually much need to transform the data to make it normal (e.g. Crow and Shimizu, 1988, Lognormal distributions, Dekker Inc, NY). > The most difficult situation is that if scientifically the relationship > between x and y is linear, should the data transformation still be carried > out (just to satisfy the statistical requirement)? When the relation between x and y is linear such as in E(y)=a*x+b, the errors are assumed to be additive (people usually do not believe that y_i=(a*x_i+b)*e_i but rather y_i=(a*x_i+b)+e_i), so applying a log transform to such case does not satisfy statistical requirements. On the contrary, it goes against statistical advice. The multiplicative error structure arises in models of the form E(y)=a*x^b or E(y)=a*exp(b*x), or in general, in all multiplicative processes. As there is a central limit theoren for additive processes leading to the normal, there is a central limit theorem for multiplicative processes leading to the lognormal. In the case of geostatistics, as pointed out by other people, the kriging equations do not require distributional assumptions (though the fitting of the model variogram to the moment-based Matheron variogram does). If the frequency distribution of the regionalised variable looks lognormal, it means that there is un underlying mechanism which is multiplicative, but still i don't see why the variable should be transformed for geostatistical analysis, except perhaps for fitting the variogram. Cheers Ruben -- * To post a message to the list, send it to [EMAIL PROTECTED] * As a general service to the users, please remember to post a summary of any useful responses to your questions. * To unsubscribe, send an email to [EMAIL PROTECTED] with no subject and "unsubscribe ai-geostats" followed by "end" on the next line in the message body. DO NOT SEND Subscribe/Unsubscribe requests to the list * Support to the list is provided at http://www.ai-geostats.org
